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July 2, 2015 page compiled on July 2, 2015 at 5:57pm

is Stationary: If all its statistics do not change with shift of origin

is Wide Sense Stationary: If the mean is constant, and

where autocorelation is deﬁned as Note, if is real, then is real and even

Note: must be WSS if it is ergodicSo ergodic process has constant mean.

input . Find which is Hilbert transform of deﬁned as

An easy way is to ﬁrst ﬁnd which is the Fourier transform of and then inverse it to ﬁnd

input: random signal

output: PSD of

Algorithm:

- 1.
- Find autocorrelation of
- 2.
- Find the Fourier Transform of . The result is the PSD of called

Another method (this below works if not random ) , why? can;t ﬁnd FT for random process?

- 1.
- Find Fourier Transform of
- 2.
- Find the

Variance is the sum of the total average normalized power and the DC power.

For the a signal whose mean is zero,

How to ﬁnd average, power, PEP, eﬀective value (or the RMS) of a periodic function?

Let be a periodic function, of period , then

The average power is

Eﬀective value, or the RMS value is

For example, for

To ﬁnd PEP (which is the peak envelope power), ﬁnd the complex envelope , then ﬁnd the average power of it. i.e.

Suppose we have a message that is sampled. Assume we have bits to use for encoding the sample levels. Hence there are levels of quantizations. We want to ﬁnd the ration of the signal to the noise power. Noise here is generated due to quantization (i.e. due to the rounding oﬀ values of during sampling).

This is the algorithm:

Input: , the number of bits for encoding, absolute maximum value of the message , the pdf of the message is is random message or function if it is deterministic (such as )

- 1.
- Find the quantization step size
- 2.
- Find of the error is where is the step size found in (1), hence
- 3.
- If is deterministic ﬁnd
- 4.
- If is random, ﬁnd , this is called the second moment of the pdf
- 5.

Hence ﬁnd for noise quantisation comes down to ﬁnding the power in the message .

Examples: For sinosoidal message , . For random with PDF which is uniform distributed , for random which is AWGN. Do this later

Given an analog value say and given a maximum absolute possible value to be , and given the number of bits available for coding to be , the following are the algorithm to generate the quantiazed version of , called

Input:

output:

Let called the step size

Let which is the quantization level

If then end if

if then else endif

Input:

output:

Let called the step size

Let which is the quantization level

If then end if

If then else endif

Input:

output:

Let called the step size

Let which is the quantization level

If then

if then

end if

else

if then

end if

end if

Input:

output:

Let called the step size

Let which is the quantization level

If then

if then

end if

else

if then

end if

end if

For any bandpass signal, we can write it as

Where is the complex envelope of . For PM and FM, the baseband modulated signal, has the form Hence the above becomes

But , hence the above becomes

| (1) |

The above is the general form for PM and FM. Now, for PM, and for FM, . Hence, substituting in (1) we obtain

and

From the general form for angle modulated signal (see above note)

The phase deviation is . And the maximum phase deviation is simply the maximum of

Now, to ﬁnd the frequency deviation, we need a little bit more work. Start with

Where is the instantaneous frequency in Hz. But

First ﬁnd , for to ﬁnd use the following

, where is the transmission bandwidth, and is the baseband bandwidth. For , . For , . For ,

Figure of merit, is deﬁned as where is the signal-to-noise ratio on output from modulator, and is signal-to-noise ratio for the channel, assuming channel has AWGN added. The following diagram shows the calculations. I used a coherent demodulator.

Now assuming , the above simpliﬁes to

Hence

Now ﬁnd

Now, to ﬁnd , which is the envelope of

Now, assuming and , then the above simpliﬁes to

now apply the DC blocker, we obtain

We notice, that for Large , this detector gives the same result as coherent detector.

For small , it is better to use the coherent detector than the envelope detector.

The diﬀerence here is that SSB signal has transmission bandwidth and not as in all the previous signals. Assume we are working with upper sideband. Analysis is the same for lower sideband.

Where is a constant. Usually but we will leave it as for now. is the Hilbert transform of

Assume , we obtain

Hence

After low pass ﬁlter, we obtain

Hence,

Hence

Hence

is the output of VSB ﬁlter when input is